This article introduces the large portfolio selection using gross-exposure constraints. It shows that with gross-exposure constraints, the empirically selected optimal portfolios based on estimated covariance matrices have similar performance to the theoretical optimal ones and there is no error accumulation effect from estimation of vast covariance matrices. This gives theoretical justification to the empirical results by Jagannathan and Ma. It also shows that the no-short-sale portfolio can be improved by allowing some short positions. The applications to portfolio selection, tracking, and improvements are also addressed. The utility of our new approach is illustrated by simulation and empirical studies on the 100 FamaFrench industrial portfolios and the 600 stocks randomly selected from Russell 3000.

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We call a finite set ${\mathcal{D}}\subset {\Bbb Z}^s$ a {\it (self-affine) tile digit set} with respect to an expanding integral matrix ${\bf A}$ if the self-affine set $T({\bf A}, \D)$ is a tile in ${\Bbb R}^s$. It has been a widely open problem to characterize the tile digit sets for a given ${\bf A}$. While there are substantial investigations on ${\Bbb R}$, there is no result on ${\Bbb R}^s$ other than the case where $|\det {\bf A}| =p$ with $p$ a prime. In this paper, we make an initiation to study a basic case ${\bf A} = p{\bf I}_2$ in ${\Bbb R}^2$. We characterize the tile digit sets by making use of the zeros of the mask polynomial of ${\mathcal{D}}$ associated with a tile criterion of Kenyon [K], together with a recent result of Iosevich {\it et al} on factorization of sets in ${\Bbb Z}_p \times {\Bbb Z}_p$ [IMP].

Blocking is commonly used in randomized experiments to increase efficiency of estimation. A generalization of blocking removes allocations with imbalance in covariate distributions between treated and control units, and then randomizes within the remaining set of allocations with balance. This idea of rerandomization was formalized by Morgan and Rubin (Annals of Statistics, 2012, 40, 1263–1282), who suggested using Mahalanobis distance between treated and control covariate means as the criterion for removing unbalanced allocations. Kallus (Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2018, 80, 85–112) proposed reducing the set of balanced allocations to the minimum. Here we discuss the implication of such an ‘optimal’ rerandomization design for inferences to the units in the sample and to the population from which the units in the sample were randomly drawn. We argue that, in general, it is a bad idea to seek the optimal design for an inference because that inference typically only reflects uncertainty from the random sampling of units, which is usually hypothetical, and not the randomization of units to treatment versus control.

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.